Independence

Events are independent when they are not related to each other; that is, the outcome of one has no bearing on the outcome of another.

Suppose we have two independent events, A and B. Then, we can test the following:

If this is not true, then the events are dependent. 

Imagine you're at a casino and you're playing craps. You throw two dice—their outcomes are independent of each other. 

An interesting property of independence is that if A and B are independent events, then so are A and B^C. 

Let's take a look and see how this works:

When we have multiple events, A₁, A₂, …, A_n, we call them mutually independent when  for all cases of n ≥ 2. 

Let's suppose we conduct two experiments in a lab; we model them independently as  and  and the probabilities of each are  and , respectively. If the two are independent, then we have the following:

This is for all cases of i and j, and our new sample space is Ω = Ω₁ × Ω₂. 

Now, say A and B are events in the Ω₁ and Ω₂ experiments, respectively. We can view them as subspaces of the new sample space, Ω, by calculating A × Ω₂ and B × Ω₁, which leads to the following:

Even though we normally define independence as different (unrelated) results in the same experiment, we can extend this to an arbitrary number of independent experiments as well.